scholarly journals Commuting pairs of endomorphisms of

2016 ◽  
Vol 38 (3) ◽  
pp. 1025-1047
Author(s):  
LUCAS KAUFMANN
Keyword(s):  
Homogeneous Polynomial ◽  
Polynomial Maps ◽  
Disjoint Sequence ◽  
Commuting Pairs ◽  
Number Of Iterations

We consider commuting pairs of holomorphic endomorphisms of $\mathbb{P}^{2}$ with disjoint sequence of iterates. The case that has not been completely studied is when their degrees coincide after some number of iterations. We show in this case that they are either commuting Lattès maps or commuting homogeneous polynomial maps of $\mathbb{C}^{2}$ inducing a Lattès map on the line at infinity.

scholarly journals CLASSIFICATION OF CUBIC HOMOGENEOUS POLYNOMIAL MAPS WITH JACOBIAN MATRICES OF RANK TWO

2018 ◽  
Vol 98 (1) ◽  
pp. 89-101 ◽  
Author(s):  
MICHIEL DE BONDT ◽  
XIAOSONG SUN
Keyword(s):  
Homogeneous Polynomial ◽  
Jacobian Matrix ◽  
Jacobian Matrices ◽  
Polynomial Maps ◽  
Keller Map

Let $K$ be any field with $\text{char}\,K\neq 2,3$. We classify all cubic homogeneous polynomial maps $H$ over $K$ whose Jacobian matrix, ${\mathcal{J}}H$, has $\text{rk}\,{\mathcal{J}}H\leq 2$. In particular, we show that, for such an $H$, if $F=x+H$ is a Keller map, then $F$ is invertible and furthermore $F$ is tame if the dimension $n\neq 4$.

On the Strong Nilpotence Problem

2014 ◽  
Vol 21 (01) ◽  
pp. 117-128 ◽  
Author(s):  
Xiaosong Sun
Keyword(s):  
Homogeneous Polynomial ◽  
Polynomial Maps ◽  
Strongly Nilpotent

In this paper, we describe the structure of quadratic homogeneous polynomial maps F=X+H with JH3=0. As a consequence we show that in dimension n ≤ 6, JH is strongly nilpotent, or equivalently F=X+H is linearly triangularizable.

scholarly journals A remark about homogeneous polynomial maps

2002 ◽  
Vol 19 (2) ◽  
pp. 257 ◽  
Author(s):  
Alexey A. Tret'yakov ◽  
Henryk Żołądek
Keyword(s):  
Homogeneous Polynomial ◽  
Polynomial Maps

Ont he monodromy representation of polynomial maps in n variables

2002 ◽  
Vol 39 (3-4) ◽  
pp. 361-367
Author(s):  
A. Némethi ◽  
I. Sigray
Keyword(s):  
Cohomology Group ◽  
Jacobian Conjecture ◽  
Natural Basis ◽  
Generic Fiber ◽  
Monodromy Representation ◽  
Polynomial Maps ◽  
Polynomial Map

For a   non-constant polynomial map f: Cn?Cn-1 we consider the monodromy representation on the cohomology group of its generic fiber. The main result of the paper determines its dimension and provides a natural basis for it. This generalizes the corresponding results of [2] or [10], where the case n=2 is solved. As  applications,  we verify the Jacobian conjecture for (f,g) when the generic fiber of f is either rational or elliptic. These are generalizations of the corresponding results of [5], [7], [8], [11] and [12], where the case  n=2 is treated.

scholarly journals A Note on the Decomposition Methods for Support Vector Regression

2002 ◽  
Vol 14 (6) ◽  
pp. 1267-1281 ◽  
Author(s):  
Shuo-Peng Liao ◽  
Hsuan-Tien Lin ◽  
Chih-Jen Lin
Keyword(s):  
Support Vector Regression ◽  
Decomposition Method ◽  
Decomposition Methods ◽  
Support Vector ◽  
Dual Formulation ◽  
Working Set ◽  
Number Of Iterations

The dual formulation of support vector regression involves two closely related sets of variables. When the decomposition method is used, many existing approaches use pairs of indices from these two sets as the working set. Basically, they select a base set first and then expand it so all indices are pairs. This makes the implementation different from that for support vector classification. In addition, a larger optimization subproblem has to be solved in each iteration. We provide theoretical proofs and conduct experiments to show that using the base set as the working set leads to similar convergence (number of iterations). Therefore, by using a smaller working set while keeping a similar number of iterations, the program can be simpler and more efficient.

scholarly journals On the Locally Polynomial Complexity of the Projection-Gradient Method for Solving Piecewise Quadratic Optimisation Problems

Entropy ◽  
2021 ◽  
Vol 23 (4) ◽  
pp. 465
Author(s):  
Agnieszka Prusińska ◽  
Krzysztof Szkatuła ◽  
Alexey Tret’yakov
Keyword(s):  
Computational Complexity ◽  
Polynomial Complexity ◽  
Initial Point ◽  
Quadratic Functions ◽  
Problem Dimension ◽  
Piecewise Quadratic Functions ◽  
Large Systems ◽  
Systems Of Linear Inequalities ◽  
Number Of Iterations ◽  
Finite Number Of Iterations

This paper proposes a method for solving optimisation problems involving piecewise quadratic functions. The method provides a solution in a finite number of iterations, and the computational complexity of the proposed method is locally polynomial of the problem dimension, i.e., if the initial point belongs to the sufficiently small neighbourhood of the solution set. Proposed method could be applied for solving large systems of linear inequalities.

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